 This is the equation of motion in the horizontal plane, in vertical plane because there is no bending so 1 by rho 0 square term which was here which shows the bending of the trajectory will not be there and only gradient related term will be there in the vertical plane. So in vertical plane we got the equation of motion like this, now you can see that this quantity can be written down as k y. So our equation of motion will be d 2 s by d s square plus k y y is equal to 0. Similarly this quantity can be written as k x. So our equation of motion will be d 2 x by d s square plus dx y is equal to 0. So these equations are similar to simple harmonic oscillator equation in horizontal and in vertical plane. k x is here 1 by rho square plus 1 by b 2 0 del b by del x and in the case of vertical plane it is minus 1 by b rho 0 del b by del x, this y should not be there. Here you can see that again b rho 0 appeared in the denominator in k x and k y also means these strength are normalized by the momentum and so our optics description using these strength become momentum independent. Suppose we are talking about the 2 g v beam, we used some k x k y value for that and when accelerator is tuned to have 4 g v beam then again same k x k y will be used because it is momentum independent description. When we will put the value of b rho 0 corresponding to 2 g v or 4 g v we will get the required gradient for that energy. Now you can see certain features of this equation of motion, first of all we use Maxwell's Curl equation in the magnetic field. This will give you del b y by del x minus del b x by del y is equal to 0 and s b s is 0 so that component will not appear. So del b y by del x minus del b y by del x is equal to 0 means del b y by del x is equal to del b x by del y means gradient in the horizontal plane will always be equal to the gradient in the vertical plane. So this del b y by del x will always be equal to del b y by del x. This is the requirements which will satisfy the Maxwell's equation. If magnetic field decreases with radius del b y by del x is negative then definitely k y will be positive. How we can see here that suppose magnetic pole faces are like this this is the x. So here magnetic field lines are in perfectly perpendicular direction here at the center. So b x is 0 here and now as magnetic fields is become weaker and weaker in the x plane the x component also changes b x and b x is here larger. So here p y is becoming weaker and b x is becoming larger means if k x is positive k y will be negative if k y will be positive k x will be negative. So this k x and k y under this k x and k y this equation of motion is like the simple harmonic oscillations means there are some kind of bounded motion of the particles and bounded motion means some kind of focusing means particle is not going away from the trajectory this is oscillating around the trajectory means it is bounded around the trajectory means this is a kind of focusing. So k x and k y shows the focusing action. If in complete optics k y become negative or k x become negative means no bounded motion will be clear and these will be exponential solution which is growing means distance from the design axis of that particle trajectory will increase and increase. So that is you can see de-focusing action. Suppose gradient is 0 in this equation still you have k x is equal to 1 by 2 0 square means still you have some positive x in the horizontal even if gradient is not here. So this shows that even in the absence of gradient there is some kind of focusing in the dipole magnet in the horizontal plane. We will see a bit later that what is this kind of focusing this is known as geometrical focusing. Often this gradient is written down in this way a field index or a new parameter n is introduced in which minus rho 0 by b 0 del b y by del x is written as n and using this equation using this parameter above equation can be written down as for horizontal plane d 2 x by d s square plus 1 minus n x by 2 0 square and for vertical plane it will be d 2 y by d s square plus n by 2 0 square. Now here you can see that if n is greater than 1 this quantity in the bracket will become negative and in horizontal plane there will be de-focusing. So n should be less than 1 for focusing in the horizontal plane and here you can see that if n cannot have a negative value if n is negative then de-focusing will occur in the vertical plane. So in vertical plane if you want focusing n should be greater than 0. So for horizontal plane n should be less than 1 and for vertical plane n should be greater than 0 for focusing action. So n should be between the value 0 to 1 and in this range we will get focusing in both the planes. Now what is geometric focusing? Consider there is a dipole magnet and under that dipole magnetic field a particle trajectory is making a circular path. This is our design trajectory. Now suppose a particle is deviated initially from this design trajectory here this is shown here. Now how the motion of this particle will be evolved? So definitely because this is a constant magnetic field it will evolve its own trajectory which will be circular with deviated center. So now you can see that here this deviated particle is coming closer to the design trajectory. Here it meets the design trajectory then again it goes away and again it comes here and again meet here. Both the trajectories meet here. So this deviated particle when it meets the design trajectory here it is the first focal point and you can say that this is another focal point where both trajectories meet each other. If we unfold this trajectory means if we make a straight line design trajectory means we unfold it then how the deviated particles trajectory will look like? These deviated particle trajectories will look like this. Here it is coming towards the design trajectory here it meets the design trajectory then goes away then again comes here and then again it meets here with the design trajectory. So deviated particle makes certain kind of oscillatory motion around the design trajectory. This is a boundary motion and this shows that focusing is taking place. Instead of a full circular path if we take only a sector of the circuit means our magnet has only this length. Means the design trajectory comes here it is a straight because there is no magnetic field now it enters into the magnetic field so it bends and again it exits from the design trajectory. Still we can understand the geometric focusing like this. Suppose now this is a deviated particle and coming parallel to the design trajectory it is deviated from here this is deviation from the design trajectory. When it enters in the dipolar magnetic field it covers larger path because it is on the larger side of the magnet so it covers a larger path inside the magnetic field and if it covers larger side larger path inside the magnetic field compared to the design path means it has larger bending angle. This theta 1 is the bending angle for this trajectory and this theta 0 is the bending angle for the design trajectory. So theta 1 will be larger than theta 3 means after exiting of the dipolar magnet this trajectory will come towards the design trajectory and now suppose particle is deviated in the negative x direction then it will have shorter path lengths inside the magnet. When shorter path length means this theta 2 its bending will be less compared to the design trajectory's bending so again it will go towards the design trajectory. So either the particle deviated in plus x or is it deviated in the minus x both of these particles goes towards go towards the design trajectory after exiting the magnet means some kind of focusing action is there. So whether we are taking the full circular path or sector of that circular path we can understand geometric focusing in this way and 1 by rho square term in the equation of motion in the kx was showing this geometrical focusing quantitatively. Now we are talking about the dipole magnets, quadrupole magnets, gradient etc. How really dipole magnet looks like? So a dipole magnet is made by a yoke because generally we use electromagnets in the accelerator so we can tune by changing the current so we can tune to different magnetic fields by changing the current in the coils. So this is the yoke photomagnetic material. Here is the design path from the trajectory design path from the magnet it is the trajectory and around this design path there are vacuum chamber inside which particle moves. Now there are the 4 coils so current flow like this in this coil here you can see the side plane of this magnet and when current flows through the coil magnetic field lens looks like this so this is the north hole this is the south hole by reversing the current direction we can change the polarity of the poles. Another type of design also exists for the dipole magnet here we are saying that yoke is making some c-type structure so this is a c-type magnet here yoke is making some h-type structure so this is a h-type magnet and this is a window frame. So in window frame magnet you can see that symmetry exists everywhere so this is the uniformity of the field will be best in this case and uniformity will be inferior in the c-type magnet because there is no symmetry around in the yoke. This is the actual photograph of a dipole magnet in a synchrotron. This dipole is of Indus II accelerator Indus II synchrotron radiation source situated in Arrakat Indus. So you can see that this is the yoke and these are the coils and inside this gap there will be field and we will put a vacuum chamber passing through this gap in which particles will travel. So again the references are seen as of their real vectors and in next lecture we will talk more about some other kind of focusing that is the strong focusing and some age